The source of artifacts is equation (4), which limits the interval of existence for the variable kh. Normally the offset-wavenumber kh is evenly sampled between the values in the DSR offset-midpoint equation, necessary for the FFT along the offset axis. However, the requirement for the square-root expressions to be real limits the available values for kh. Let's examine equation (4) again:
When only a subset of the values of kh are used from the whole set of discrete values, artifacts are introduced in the phase expression. Consider the form of the DSR equation presented in equation (8):
where the integral in kh has the integration limits
Equation (9) is used to migrate a single impulse in a constant-offset section with h=190m, using constant sampling in kh for all the values . In Figure the only difference between the two migration programs is the offset spacing. Bigger sampling in offset implies smaller sampling in offset-wavenumber. As a result more offset-wavenumber samples are used to compute the phase and reduce the artifacts. The ideal algorithm will use all the available samples, dividing the integration limits in equation (9) by the total number of offsets (assumed to be equal to the number of offset-wavenumbers). This will not preclude the use of FFT to transform the phase from offset-wavenumber domain to offset domain, because for each pair of values , the variable kh is evenly sampled in the domain of definition.
In order to use the different results of DSR migration algorithms I create a prestack model consisting of 64 constant-offset sections over three vertical diffractors in a depth variable velocity medium. The velocity model is shown in Figure a while an example of the zero-offset travel-time map is shown in Figure b. Figure a shows a zero-offset common midpoint (CMP) section, while Figure b shows the CMP section corresponding to the longest offset (h=630m).
The prestack model was migrated using several variations of the DSR migration algorithm. Figures a and b compare the results of DSR migration via the classic algorithm implementing equation (1) and the results of DSR migration for separate offsets implementing equation (8) using a FFT to transform the phase. In both cases kh was constantly sampled. To obtain Figure b all the separate constant-offset sections were stacked in the end. The two results are identical as is expected.
Figure a represents the results of DSR migration by implementing equation (8) directly. The integral in kh is calculated numerically for each offset. The algorithm is slower than the one evaluating the integral in kh via FFT, but it does not require the migration of all the separate constant-offset sections at the same time. Figure b represents the results of DSR migration with different sampling in kh for each pair. Because stacking attenuates the artifacts the difference between the two algorithms is not as visible in the migrated CMP section.
Figure compares the migrated results of the same CMP, for different offsets. The figure was obtained by slicing vertically the migrated data set. The slice passes through the location of the three diffraction events shown in Figure . A correctly migrated image should show the same events for all the offsets. Figure a displays the diagonal artifacts that appear using constant kh sampling. The diagonal stripes create the ghost ellipse in Figure a. Figure b is obtained using separate sampling of kh for each pair, and all artifacts are virtually eliminated.